Yes, you can do this. The way this works?

If the onclient click function returns true, then the server side code/event runs.

If the onclient click function returns false, then the server side code/event does NOT run. So you can do this:

   <asp:Button Text="search" ID="search" OnClick="search_Click"
    runat="server" style="float:right" onClientClick="return neki();"

And now your JS code can be this:

function neki() {
    if ( $("input[id*='txtIDCriteria']").val() == ''
        && $("input*[id*='chkID']").is(':checked'))
    {
    alert('Value is required.');
    return true;
    }
    return false;
}

While this will work for above, if you use a jQuery.UI dialog, the code does not wait, but there is a simple work around for even jQuery.UI dialog that respond to say a yes/no answer, and you can still conditional run the server side button code.

edit: opps - I have this backwards.

the code should be like this:

function neki() {
    if ( $("input[id*='txtIDCriteria']").val() == ''
        && $("input*[id*='chkID']").is(':checked'))
    {
    alert('Value is required.');
    return false;  <--- return false to NOT run button click
    }
    return true;   <--- return true - this will allow button click to run
}